# LCM of Algebraic Expressions ## LCM of Algebraic Expressions

To calculate the LCM (Lowest Common Multiple) of the given algebraic expressions, we should convert the algebraic expressions into their simplest factors. And, then the product of their common factors and the remaining factors will give the LCM of the given algebraic expressions.

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LCM = Common factors × Remaining factors

For example: x2 + 5x + 6 and x2 + 3x + 2 are two algenraic expressions. x2 + 5x + 6 = (x + 2)(x + 3) and x2 + 3x + 2 = (x + 1)(x+2) are their simplest factors.  And, (x + 1)(x + 2)(x + 3) is the product of their common and remaining factors. So, LCM of x2 + 5x + 6 and x2 + 3x + 2 is (x + 1)(x + 2)(x + 3). While finding LCM, we should use the following steps:

Steps:

1.     Factorize the given algebraic expressions.

2.     Take out the common factors and then the remaining factors.

3.     The product of common and remaining factors will give the required LCM.

The concept of LCM will be more clear from the following worked-out examples.

Worked Out Examples

Example 1: Find the LCM of 3x2yz, 4y2z and 5xz2

Solution:

Here,

1st expression = 3x2yz = 3 × x × x × y × z

2nd expression = 4y2z = 2 × 2 × y × y × z

3rd expression = 5xz2 = 5 × x × z × z

LCM = 3 × 2 × 2 × 5 × x × x × y × y × z × z

= 60x2y2z2

Example 2: Find the LCM of m4 – 4m2 and 3m2 + 6m

Solution:

Here,

1st expression = m4 – 4m2

= m2(m2 – 4)

= m2(m2 – 22)

= m × m(m + 2)(m – 2)

2nd expression = 3m2 + 6m

= 3m(m + 2)

LCM = 3m × m(m + 2)(m – 2)

= 3m2(m + 2)(m – 2)

Example 3: Find the LCM of a2 – 3a + 2, a4 + a3 – 6a2 and a3 + 2a2 – 3a

Solution:

Here,

1st expression = a2 – 3a + 2

= a2 – (2 + 1)a + 2

= a2 – 2a – a + 2

= a(a – 2) – 1(a – 2)

= (a – 2)(a – 1)

2nd expression = a4 + a3 – 6a2

= a2(a2 + a – 6)

= a2{a2 + (3 – 2)a – 6}

= a2(a2 + 3a – 2a – 6)

= a2{a(a + 3) – 2(a + 3)}

= a × a(a + 3)(a – 2)

3rd expression = a3 + 2a2 – 3a

= a(a2 + 2a – 3)

= a{a2 + (3 – 1)a – 3}

= a(a2 + 3a – a – 3)

= a{a(a + 3) – 1(a + 3)}

= a(a + 3)(a – 1)

LCM = a × a(a – 1)(a – 2)(a + 3)

= a2(a – 1)(a – 2)(a + 3)

Example 4: Find the LCM of x2 – 5x + 6, x2 + 4x – 12 and x2 – 2x

Solution:

Here,

1st expression = x2 – 5x + 6

= x2 – (3 + 2)x + 6

= x2 – 3x – 2x + 6

= x(x – 3) – 2(x – 3)

= (x – 3)(x – 2)

2nd expression = x2 + 4x – 12

= x2 + (6 – 2)x – 12

= x2 + 6x – 2x - 12

= x(x + 6) – 2(x + 6)

= (x + 6)(x – 2)

3rd expression = x2 – 2x

= x(x – 2)

LCM = x(x – 2)(x – 3)(x + 6)

Example 5: Find the LCM of a2 – b2 – 2bc – c2, b2 – c2 – 2ca – a2 and c2 – a2 – 2ab – b2

Solution:

Here,

1st expression = a2 – b2 – 2bc – c2

= a2 – (b2 + 2bc + c2)

= a2 – (b + c)2

= (a + b + c)(a – b – c)

2nd expression = b2 – c2 – 2ca – a2

= b2 – (c2 + 2ca + a2)

= b2 – (c + a)2

= (b + c + a)(b – c – a)

= (a + b + c)(b – c – a)

3rd expression = c2 – a2 – 2ab – b2

= c2 – (a2 + 2ab + b2)

= c2 – (a + b)2

= (c + a + b)(c – a – b)

= (a + b + c)(c – a – b)

LCM = (a + b + c)(a – b – c)(b – c – a)(c – a – b)

Example 6: Find the LCM of a3 – 2a2b + 2ab2 – b3, a4 + b4 + a2b2 and 4a4b + 4ab4

Solution:

Here,

1st expression = a3 – 2a2b + 2ab2 – b3

= a3 – b3 – 2a2b + 2ab2

= (a – b)(a2 + ab + b2) – 2ab(a – b)

= (a – b)(a2 + ab + b2 – 2ab)

= (a – b)(a2 – ab + b2)

2nd expression = a4 + b4 + a2b2

= (a2)2 + 2a2b2 + (b2)2 – a2b2

= (a2 + b2)2 –(ab)2

= (a2 + b2 + ab)(a2 + b2 – ab)

= (a2 + ab + b2)(a2 – ab + b2)

3rd expression = 4a4b + 4ab4

= 4ab(a3 + b3)

= 4ab(a + b)(a2 – ab + b2)

LCM = 4ab(a + b)(a – b)(a2 + ab + b2)(a2 – ab + b2)

Example 7: Find the LCM of a2 – 18a – 19 + 20b – b2, a2 + a – b2 + b and 4a2 – 4b2 + 8b - 4

Solution:

Here,

1st expression = a2 – 2.a.9 + 92 – 100 + 20b – b2

= (a – 9)2 – (102 – 2.10.b + b2)

= (a – 9)2 – (10 – b)2

= {(a – 9) + (10 – b)}{(a – 9) – (10 – b)}

= (a – 9 + 10 – b)(a – 9 – 10 + b)

= (a – b + 1)(a + b – 19)

2nd expression = a2 + a – b2 + b

= a2 – b2 + a + b

= (a + b)(a – b) + 1(a + b)

= (a + b)(a – b + 1)

3rd expression = 4a2 – 4b2 + 8b - 4

= 4(a2 – b2 + 2b – 1)

= 4{a2 – (b2 – 2b + 1)}

= 4{a2 – (b – 1)2}

= 4(a + b – 1)(a – b + 1)

LCM = 4(a + b)(a + b – 1)(a – b + 1)(a + b – 19)

Example 8: Find the LCM of 1 + 4a + 4a2 – 16a4, 1 + 2a – 8a3 – 16a4 and 1 – 8a3

Solution:

Here,

1st expression = 1 + 4a + 4a2 – 16a4

= 12 + 2.1.2a + (2a)2 – (4a2)2

= (1 + 2a)2 – (4a2)2

= (1 + 2a + 4a2)(1 + 2a – 4a2)

2nd expression = 1 + 2a – 8a3 – 16a4

= 1(1 + 2a) – 8a3(1 + 2a)

= (1 + 2a)(1 – 8a3)

= (1 + 2a){13 – (2a)3}

= (1 + 2a)(1 – 2a)(1 + 2a + 4a2)

3rd expression = 1 – 8a3

= 13 – (2a)3

= (1 – 2a)(1 + 2a + 4a2)

LCM = (1 + 2a)(1 – 2a)(1 + 2a + 4a2)(1 + 2a – 4a2)

If you have any question or problems regarding the LCM of algebraic expressions, you can ask here, in the comment section below.

1. 2. find lcm of a^3+1,a^6-1 and a^4+a^2+1

1. Here is the solution to your problems,

You can solve it by yourself using Quick Math Solver. Please install the app by using the link given above. If you have more problems, post it here.

3. LCM of
a^4 + 4b^4 + (a² + b²)5b²,
a² + 2a³b + a²b² -9b^4

1. Here is the solution to your problem,

4. Very efficiently written information. It will be beneficial to anybody who utilizes it, including me. Keep up the good work. For sure i will check out more posts. This site seems to get a good amount of visitors. Quantitative Reasoning Math

5. Factorization of 4a^3-16a^2+20a-8
4acube +16 a square +20a -8

1. You can solve it by using our app Quick Math Solver. Please Install it on your android phone.
Here is the solution to your problems,